Scalable Low-Order Finite Element Preconditioners for High-Order Spectral Element Poisson Solvers

2019 ◽  
Vol 41 (5) ◽  
pp. S2-S18 ◽  
Author(s):  
Pedro D. Bello-Maldonado ◽  
Paul F. Fischer
Keyword(s):  
Finite Element ◽  
Spectral Element ◽  
High Order ◽  
Low Order

The Double Absorbing Boundary Method Incorporated in a High-Order Spectral Element Formulation

2020 ◽  
Vol 28 (04) ◽  
pp. 2050007
Author(s):  
Symeon Papadimitropoulos ◽  
Daniel Rabinovich ◽  
Dan Givoli
Keyword(s):  
Numerical Solution ◽  
Spectral Element ◽  
High Order ◽  
Element Formulation ◽  
Finite Element Scheme ◽  
Absorbing Boundary ◽  
Long Time ◽  
Artificial Boundaries ◽  
Wave Problems ◽  
Low Order

In this paper, we consider the numerical solution of the time-dependent wave equation in a semi-infinite waveguide. We employ the Double Absorbing Boundary (DAB) method, by introducing two parallel artificial boundaries on the side where waves are outgoing. In contrast to the original implementation of the DAB, where the numerical solution involved either a low-order finite difference scheme or a low-order finite element scheme, here we incorporate the DAB into a high-order spectral element formulation, which provides us with very accurate solutions of wave problems in unbounded domains. This is demonstrated by numerical experiments. While the method is highly accurate, it suffers from long-time instability. We show how to postpone the onset of the instability by a prudent choice of the computational parameters.

scholarly journals A High-order Staggered Finite-Element Vertical Discretization for Non-Hydrostatic Atmospheric Models

2016 ◽  
Author(s):  
J. E. Guerra ◽  
P. A. Ullrich
Keyword(s):  
Finite Element ◽  
Spectral Element ◽  
Atmospheric Modeling ◽  
High Order ◽  
High Order Accuracy ◽  
Staggered Grid ◽  
Test Case ◽  
Sound Waves ◽  
Vertical Coordinate ◽  
Hydrostatic Balance

Abstract. Atmospheric modeling systems require economical methods to solve the non-hydrostatic Euler equations. Two major differences between hydrostatic models and a full non-hydrostatic description lies in the vertical velocity tendency and numerical stiffness associated with sound waves. In this work we introduce a new arbitrary-order vertical discretization entitled the Staggered Nodal Finite Element Method (SNFEM). Our method uses a generalized discrete derivative that consistently combines the Discontinuous Galerkin and Spectral Element methods on a staggered grid. Our combined method leverages the accurate wave propagation and conservation properties of spectral elements with staggered methods that eliminate stationary (2Δx) modes. Furthermore, high-order accuracy also eliminates the need for a reference state to maintain hydrostatic balance. In this work we demonstrate the use of high vertical order as a means of improving simulation quality at relatively coarse resolution. We choose a test case suite that spans the range of atmospheric flows from predominantly hydrostatic to nonlinear in the Large Eddy regime. Our results show that there is a distinct benefit in using the high-order vertical coordinate at low resolutions with the same robust properties of the low-order alternative.

scholarly journals A high-order staggered finite-element vertical discretization for non-hydrostatic atmospheric models

2016 ◽  
Vol 9 (5) ◽  
pp. 2007-2029 ◽  
Author(s):  
Jorge E. Guerra ◽  
Paul A. Ullrich
Keyword(s):  
Finite Element ◽  
Spectral Element ◽  
Atmospheric Modeling ◽  
High Order ◽  
High Order Accuracy ◽  
Staggered Grid ◽  
Test Case ◽  
Sound Waves ◽  
Vertical Coordinate ◽  
Hydrostatic Balance

Abstract. Atmospheric modeling systems require economical methods to solve the non-hydrostatic Euler equations. Two major differences between hydrostatic models and a full non-hydrostatic description lies in the vertical velocity tendency and numerical stiffness associated with sound waves. In this work we introduce a new arbitrary-order vertical discretization entitled the staggered nodal finite-element method (SNFEM). Our method uses a generalized discrete derivative that consistently combines the discontinuous Galerkin and spectral element methods on a staggered grid. Our combined method leverages the accurate wave propagation and conservation properties of spectral elements with staggered methods that eliminate stationary (2Δx) modes. Furthermore, high-order accuracy also eliminates the need for a reference state to maintain hydrostatic balance. In this work we demonstrate the use of high vertical order as a means of improving simulation quality at relatively coarse resolution. We choose a test case suite that spans the range of atmospheric flows from predominantly hydrostatic to nonlinear in the large-eddy regime. Our results show that there is a distinct benefit in using the high-order vertical coordinate at low resolutions with the same robust properties as the low-order alternative.

Hybrid Reduced Model of Rotor

2013 ◽  
Vol 60 (3) ◽  
pp. 319-333
Author(s):  
Rafał Hein ◽  
Cezary Orlikowski
Keyword(s):  
Finite Element Method ◽  
Finite Element ◽  
Hybrid Model ◽  
Rotor System ◽  
Reduced Model ◽  
Order Model ◽  
Gyroscopic Effect ◽  
Reduced Models ◽  
Rigid Finite Element Method ◽  
Low Order

Abstract In the paper, the authors describe the method of reduction of a model of rotor system. The proposed approach makes it possible to obtain a low order model including e.g. non-proportional damping or the gyroscopic effect. This method is illustrated using an example of a rotor system. First, a model of the system is built without gyroscopic and damping effects by using the rigid finite element method. Next, this model is reduced. Finally, two identical, low order, reduced models in two perpendicular planes are coupled together by means of gyroscopic and damping interaction to form one model of the system. Thus a hybrid model is obtained. The advantage of the presented method is that the number of gyroscopic and damping interactions does not affect the model range

scholarly journals A New Mixed Functional-probabilistic Approach for Finite Element Accuracy

2020 ◽  
Vol 20 (4) ◽  
pp. 799-813
Author(s):  
Joël Chaskalovic ◽  
Franck Assous
Keyword(s):  
Finite Element ◽  
Finite Elements ◽  
Asymptotic Relation ◽  
Probabilistic Approach ◽  
Random Variable ◽  
High Order ◽  
Relative Accuracy ◽  
The Difference ◽  
New Perspective

AbstractThe aim of this paper is to provide a new perspective on finite element accuracy. Starting from a geometrical reading of the Bramble–Hilbert lemma, we recall the two probabilistic laws we got in previous works that estimate the relative accuracy, considered as a random variable, between two finite elements {P_{k}} and {P_{m}} ({k<m}). Then we analyze the asymptotic relation between these two probabilistic laws when the difference {m-k} goes to infinity. New insights which qualify the relative accuracy in the case of high order finite elements are also obtained.

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